Prob Event bind le add bad of disagree
probEvent_bind_le_add_bad_of_disagree
Plain-language statement
Three-way disagreement-aware additive bind bound (hop A). A coupled three-world variant of probEvent_bind_le_add_of_disagree: the three worlds share the sampling computation mx, and at each shared sample x, off the disagreement set D the my-world is bounded by the oc-world plus the per-step slack ε, while on D the ob-world (the bad w...
Exact Lean statement
lemma probEvent_bind_le_add_bad_of_disagree {mx : m α}
{my : α → m β} {oc : α → m β} {ob : α → m γ}
{q : β → Prop} {r : γ → Prop} {D : α → Prop} {ε : ℝ≥0∞}
(hbad : ∀ x ∈ support mx, D x → Pr[ r | ob x] = 1)
(h : ∀ x ∈ support mx, ¬ D x → Pr[ q | my x] ≤ Pr[ q | oc x] + ε) :
Pr[ q | mx >>= my] ≤ Pr[ q | mx >>= oc] + Pr[ r | mx >>= ob] + εFormal artifact
Lean source
lemma probEvent_bind_le_add_bad_of_disagree {mx : m α} {my : α → m β} {oc : α → m β} {ob : α → m γ} {q : β → Prop} {r : γ → Prop} {D : α → Prop} {ε : ℝ≥0∞} (hbad : ∀ x ∈ support mx, D x → Pr[ r | ob x] = 1) (h : ∀ x ∈ support mx, ¬ D x → Pr[ q | my x] ≤ Pr[ q | oc x] + ε) : Pr[ q | mx >>= my] ≤ Pr[ q | mx >>= oc] + Pr[ r | mx >>= ob] + ε := by classical have := Classical.decPred q have := Classical.decPred r rw [probEvent_bind_eq_tsum, probEvent_bind_eq_tsum, probEvent_bind_eq_tsum] calc ∑' x, Pr[= x | mx] * Pr[q | my x] ≤ ∑' x, (Pr[= x | mx] * Pr[q | oc x] + Pr[= x | mx] * Pr[r | ob x] + Pr[= x | mx] * ε) := by refine ENNReal.tsum_le_tsum fun x => ?_ by_cases hx : x ∈ support mx · by_cases hDx : D x · exact ((mul_le_mul' le_rfl probEvent_le_one).trans_eq (by rw [hbad x hx hDx])).trans (le_add_right (le_add_left le_rfl)) · exact ((mul_le_mul' le_rfl (h x hx hDx)).trans_eq (left_distrib ..)).trans (add_le_add (le_add_right le_rfl) le_rfl) · simp [probOutput_eq_zero_of_not_mem_support hx] _ = (∑' x, Pr[= x | mx] * Pr[q | oc x]) + (∑' x, Pr[= x | mx] * Pr[r | ob x]) + (∑' x, Pr[= x | mx] * ε) := by rw [ENNReal.tsum_add, ENNReal.tsum_add] _ ≤ (∑' x, Pr[= x | mx] * Pr[q | oc x]) + (∑' x, Pr[= x | mx] * Pr[r | ob x]) + ε := by refine add_le_add le_rfl ?_ rw [ENNReal.tsum_mul_right] exact mul_le_of_le_one_left zero_le tsum_probOutput_le_one- Project
- VCVio
- License
- Apache-2.0
- Commit
- 2ceb2d825ee3
- Source
- VCVio/EvalDist/Monad/Disagreement.lean:71-99
Reuse this declaration
Bring the exact result into your workflow
The import identifies the source module. Your project still needs the pinned package dependency shown on this page.
What this badge means
This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.
Continue in this project
Related declarations
Expected Cost Nat eq sum tail probs of pathwise Cost At Most
AddWriterT.expectedCostNat_eq_sum_tail_probs_of_pathwiseCostAtMost
Plain-language statement
Finite tail-sum formula for natural-valued writer cost under a pathwise upper bound. If every execution path of oa incurs cost at most n, then the tail probabilities vanish above n, so the infinite tail sum truncates to Finset.range n.
Source project: VCVio
Person-level attribution pending.
IND CPA advantage to Real le sum step signed Advantage Real abs
AsymmEncAlg.IND_CPA_advantage_toReal_le_sum_step_signedAdvantageReal_abs
Plain-language statement
Planned generic one-time-to-many-time lift: bounded multi-query IND-CPA advantage is at most the sum of the extracted one-time signed advantages over the first q fresh LR queries.
Source project: VCVio
Person-level attribution pending.
IND CPA LR hybrid Game q eval Dist eq left of Makes At Most Queries
AsymmEncAlg.IND_CPA_LR_hybridGame_q_evalDist_eq_left_of_MakesAtMostQueries
Plain-language statement
If an adversary makes at most q fresh LR queries, then the leftUntil = q LR-hybrid is the all-left endpoint game.
Source project: VCVio
Person-level attribution pending.